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In this section we give the complete formulation for TB model ...

  1. 1. RSS Tech. Proposal 121599A-1 Revised: November 2, 2000 Algorithm Theoretical Basis Document (ATBD) Version 2 AMSR Ocean Algorithm Principal Investigator: Frank J. Wentz Co-Investigator: Thomas Meissner Prepared for: EOS Project Goddard Space Flight Center National Aeronautics and Space Administration Greenbelt, MD 20771 Prepared by:
  2. 2. 2
  3. 3. Table of Contents 1. Overview and Background Information 1 1.1. Introduction 1 1.2. Objectives of Investigation 2 1.3. Approach to Algorithm Development 2 1.4. Algorithm Development Plan 3 1.5. Concerns Regarding Sea-Surface Temperature Retrieval 3 1.6. Historical Perspective 5 1.7. AMSR Instrument Characteristics 7 2. Geophysical Model for the Ocean and Atmosphere 10 2.1. Introduction 10 2.2. Radiative Transfer Equation 10 2.3. Model for the Atmosphere 13 2.4. Dielectric Constant of Sea-Water and the Specular Sea Surface 20 2.5. The Wind-Roughened Sea Surface 23 2.6. Atmospheric Radiation Scattered by the Sea Surface 28 2.7. Wind Direction Effects 29 3. The Ocean Retrieval Algorithm 31 3.1. Introduction 31 3.2. Multiple Linear Regression Algorithm 31 3.3. Derivation and Testing of the Linear Regression Algorithm 32 3.4. Non-Linear, Iterative Algorithm 36 3.5. Post-Launch In-Situ Regression Algorithm 38 3.6. Incidence Angle Variations 39 4. Level-2 Data Processing Issues 40 4.1. Retrievals at Different Spatial Resolutions 40 4.2. Granules and Metadata 43 4.3. Requirements for Ancillary Data Sets 44 4.4. Computer Resources and Programming Standards 45 5. Validation for the Ocean Products Suite 46 5.1. Introduction 46 5.2. Sea-Surface Temperature Validation 47 5.3. Wind Speed Validation 49 5.4. Water Vapor Validation 50 5.5. Cloud Water Validation 52 6. References 55 List of Figures i
  4. 4. 1. Development steps for ocean algorithm 4 2. Block diagram for PM AMSR feedhorns and radiometers 9 3. The atmospheric absorption spectrum for oxygen, water vapor,… 15 4. The effective air temperature TD for downwelling radiation… 17 5. Derivation and testing of the linear regression algorithm 33 6. Preliminary results for the linear statistical regression algorithm… 35 7. Data processing flow for ocean algorithm 41 8. Locations of data buoys 49 9. Radiosonde stations on small islands 52 10. Probability density functions (pdf) for liquid cloud water 54 List of Tables 1. Expected retrieval accuracy for the ocean products 1 2. Comparison of past and future satellite radiometer systems 2 3. Instrument specifications for PM AMSR 8 4. Model coefficients for the atmosphere 18 5. RMS error in oxygen and water vapor absorption approx… 18 6. Coefficients for rayleigh absorption 20 7. Model coefficients for geometric optics 27 8. The coefficients m1 and m2 … 28 9. Preliminary estimate of retrieval error 36 10. AMSR level-2 ocean data record 43 11. Ancillary data sets required by level-2 ocean algorithm 44 12. Some of the available SST products 47 13. NDBC moored buoy open water locations as of July 1996 48 14. Island radiosonde station locations as of September 1996 51 ii
  5. 5. List of Symbols Symbol Definition Units aO1, aO2 coefficients for oxygen absorption see Table 4 aV1, aV2 coefficients for water vapor absorption see Table 4 aL1, aL2 coefficients for liquid water absorption see Table 6 A the A-matrix relating Y to X arbitrary AO vertically integrated oxygen absorption naper AV vertically integrated water vapor absorption naper AL vertically integrated cloud liquid water absorption naper b0 to b7 coefficients for effective air temperature see Table 4 c speed of light cm/s C chlorinity of sea water parts/thousand E sea-surface emissivity none f fractional foam coverage none F foam+diffraction factor for sea-surface reflectivity none h Planck’s constant in eq. (2) erg-s h height above Earth surface, elsewhere cm h0 surface roughness length cm hi, hs h-pol vectors for incident and scattered radiation none Iλ specific intensity erg/s-cm3-ster j −1 none k Boltzmann’s constant erg/K ki upward unit propagation vector none ks downward unit propagation vector none L vertically integrated cloud liquid water mm m1 , m2 coefficients for foam+diffraction factor s/m n unit normal vector for tilted surface facet none P column vector of geophysical parameters varies P(Su,Sc) probability density function of surface slope none Pλ specific power erg/s p unit polarization vector none iii
  6. 6. Symbol Definition Units r0 to r3 coefficients for geometric optics see Table 7 R total sea-surface reflectivity none R0 specular reflectivity none Rclear foam-free sea-surface reflectivity none Rgeo geometric optics sea-surface reflectivity none R× reflectivity of secondary intersection none s path length in Section 2.2 cm s salinity, elsewhere parts/thousand S total path length through atmosphere cm Sc crosswind slope for tilted surface facet none Su upwind slope for tilted surface facet none tS sea-surface temperature Celsius T temperature K TB brightness temperature K TBU upwelling atmospheric brightness temperature K TBD downwelling atmospheric brightness temperature K TBΩ sky radiation scattered upward by Earth surface K TB↑ upwelling surface brightness temperature K TB↓ downwelling cold space brightness temperature K TC cold space brightness temperature K TD effective temperature for downwelling radiation K TE effective temperature of surface+atmosphere K TS sea-surface temperature K TU effective temperature for upwelling radiation K TV typical sea temperature for given water vapor K vi, vs v-pol vectors for incident and scattered radiation none V vertically integrated water vapor mm W wind speed 10 m above sea surface m/s X column input vector arbitrary Y column output vector arbitrary ~ measurement vector arbitrary Y iv
  7. 7. Symbol Definition Units α total absorption coefficient naper/cm αO oxygen absorption coefficient naper/cm αV water vapor absorption coefficient naper/cm αL cloud liquid water absorption coefficient naper/cm β diffraction factor for sea-surface reflectivity none γ coefficients for wind direction variation of E none ∆S2 total slope variance of sea surface none ε TB measurement error in Section 3 K ε complex dielectric constant of water, elsewhere none ε∞ dielectric constant at infinite frequency none εS static dielectric constant of sea water none εS0 static dielectric constant of distilled water none εφ error in specifying wind direction degree εTs error in specifying sea-surface temperature K θi incidence angle degree θs zenith angle degree κ reduction in sea-surface reflectivity due to foam none ϕi azimuth angle for ki degree ϕs azimuth angle for ks degree φ wind direction relative to azimuth look direction degree λ radiation wavelength cm λR relaxation wavelength of sea water cm λR0 relaxation wavelength of distilled water cm µ change in TB w.r.t. incidence angle K/degree η spread factor for relaxation wavelengths none Ω fit parameter for sea surface scattering integral none Ξ error correlation matrix arbitrary ρh h-pol Frensel reflection coefficient none ρv v-pol Frensel reflection coefficient none ρL liquid water density g/cm3 ρV water vapor density g/cm3 v
  8. 8. Symbol Definition Units ρo density of water g/cm3 σ ionic conductivity of sea water s-1 σo,c co-pol. normalized radar cross section none σo,× cross-pol. normalized radar cross section none τ atmospheric transmission none ν radiation frequency GHz χ shadowing function none n correction for effective air temperature K ℑ linearizing function for TB’s none ℜ linearizing function for geophysical parameters none vi
  9. 9. 1. Overview and Background Information 1.1. Introduction With the advent of well-calibrated satellite microwave radiometers, it is now possible to obtain long time series of geophysical parameters that are important for studying the global hydrologic cycle and the Earth's radiation budget. Over the world's oceans, these radiome- ters simultaneously measure profiles of air temperature and the three phases of atmospheric water (vapor, liquid, and ice). In addition, surface parameters such as the near-surface wind speed, the sea-surface temperature, and the sea ice type and concentration can be retrieved. A wide variety of hydrological and radiative processes can be studied with these measure- ments, including air-sea and air-ice interactions (i.e., the latent and sensible heat fluxes, fresh water flux, and surface stress) and the effect of clouds on radiative fluxes. The microwave radiometer is truly a unique and valuable tool for studying our planet. This Algorithm Theoretical Basis Document (ATBD) focuses on the Advanced Mi- crowave Scanning Radiometer (AMSR) that is scheduled to fly in December 2000 on the NASA EOS-PM1 platform. AMSR will measure the Earth’s radiation over the spectral range from 7 to 90 GHz. Over the world’s oceans, it will be possible to retrieve the four im- portant geophysical parameters listed in Table 1. The rms accuracies given in Table 1 come from past investigations and on-going simulations that will be discussed. Rainfall can also be retrieved, which is discussed in a separate AMSR ATBD. We are confident that the expected retrieval accuracies for wind, vapor, and cloud will be achieved. The Special Sensor Microwave Image (SSM/I) and the TRMM microwave imager (TMI) have already demonstrated that these accuracies can be obtained. The AMSR wind retrievals will probably be more accurate than that of SSM/I and less affected by atmospher- ic moisture. A comparison between sea surface temperatures (SST) from TMI with buoy measurements indicate an rms accuracy between 0.5 and 0.7 K. One should keep in mind that part of the error arises from the temporal and spatial mismatch between the buoy mea- surement and the 50 km satellite footprint. Furthermore, the satellite is measuring the tem- perature at the surface the ocean (about 1 mm deep) whereas the buoy is measuring the bulk temperature near 1 m below the surface. There are still some concerns with regards to the sea-surface temperature retrieval, which are discussed in Section 1.5. This document is version 2 of the AMSR Ocean Algorithm ATBD. The primary difference between this version and the earlier version is that the emissivity model for the 10.7 GHz has been updated using data from TMI. In addition, there are several small updates to the radia- tive transfer model (RTM). Table 1. Expected Retrieval Accuracy for the Ocean Products Geophysical Parameter Rms Accuracy Sea-surface temperature TS 0.5 K Near-surface wind speed W 1.0 m/s Vertically integrated (i.e., columnar) water vapor V 1.0 mm Vertically integrated cloud liquid water L 0.02 mm 1
  10. 10. 1.2. Objectives of Investigation There are two major objectives of this investigation. The first is to develop an ocean re- trieval algorithm that will retrieve TS, W, V, and L to the accuracies specified in Table 1. These products will be of great value to the Earth science community. The second objective is to improve the radiative transfer model (RTM) for the ocean surface and non-raining at- mosphere. The 6.9 and 10.7 GHz channels on AMSR will provide new information on the RTM at low frequencies. Experience has shown that these two objectives are closely linked. A better understanding of the RTM leads to more accurate retrievals. A better understanding of the RTM also leads to new remote sensing techniques such as using radiometers to mea- sure the ocean wind vector. 1.3. Approach to Algorithm Development Radiative transfer theory provides the relationship between the Earth’s brightness tem- perature TB (K) as measured by AMSR and the geophysical parameters T S, W, V, and L. This ATBD addresses the inversion problem of finding TS, W, V, and L given TB. We place a great deal of emphasis on developing a highly accurate RTM. Most of our AMSR work thus far has been the development and refinement of the RTM. This work is now completed, and Section 2 describes the RTM in considerable detail. The importance of the RTM is underscored by the fact that AMSR frequency, polariza- tion, and incidence angle selection is not the same as previous satellite radiometers. Table 2 compares AMSR with other radiometer systems. Albeit some of the differences are small, they are still significant enough to preclude developing AMSR algorithms by simply using existing radiometer measurements. The differences in frequencies and incidence angle must be taken into account when developing AMSR algorithms. Table 2. Comparison of Past and Future Satellite Radiometer Systems Radiometer Frequencies/Polarization Inc. Angle SeaSat SMMR 6.6VH 10.7VH 18.0VH 21.0VH 37.0VH 49° Nimbus-7 SMMR 6.6VH 10.7VH 18.0VH 21.0VH 37.0VH 51° SSM/I 19.3VH 22.2V 37.0VH 85.5VH 53° TRMM TMI 10.7VH 19.3VH 21.0VH 37.0VH 85.5VH 53° PM AMSR 6.9VH 10.7VH 18.7VH 23.8VH 36.5VH 89.0VH 55° Our approach uses the existing radiometer measurements to calibrate various components of the RTM. The RTM formulation then provides the means to compute T B at any frequency in the 1-100 GHz range and at any incidence angle in the 50°-60° range. For the SSM/I fre- quencies and incidence angle, the resulting RTM is extremely accurate. It is able to repro- duce the SSM/I TB to a rms accuracy of about 0.6 K. (This figure comes from Table 3 in Wentz [1997], and represents the rms difference between the RTM and SSM/I observations after subtracting out radiometer noise and in situ inter-comparison errors.) As one moves away from the SSM/I frequencies and incidence angle, we do expect some degradation in the RTM accuracy. However, the hope is that the physics of the RTM is reliable enough so that this degradation is minimal when we interpolate/extrapolate to the AMSR configuration. 2
  11. 11. Given an accurate and reliable RTM, geophysical retrieval algorithms can be developed. We are developing in parallel two types of algorithms: the linear regression algorithm and the non-linear, iterative algorithm. Section 3 discusses each type of algorithm. For both types, the algorithm development is based on a simulation in which brightness temperatures for a wide variety of ocean scenes are produced by the RTM. These simulated T B’s then serve as both a training set and a test set for the algorithms. We have tested this simulation methodology by developing algorithms for SSM/I. These SSM/I algorithms are then tested using actual measurements. The results show that the SSM/I algorithms coming from the RTM simulation have essentially the same performance as those developed directly from SSM/I measurements. These results are not surprising since the RTM was calibrated to re- produce the SSM/I TB’s. This exercise is more of a closure verification of the techniques be- ing used. Simulation results for the AMSR retrieval algorithm are given in Section 3. 1.4. Algorithm Development Plan Figure 1 shows the basic steps in developing the AMSR ocean algorithm. We are current- ly developing the version 2 algorithm which includes well-calibrated 10.7-GHz ocean obser- vations from TMI. The recent TMI results show T S can be accurately retrieved in warm water above 15°C. We expect even better performance from AMSR because of the additional 6.9 GHz channel, which provides TS sensitivity in cold water. One concern is the variation of the 6.9 and 10.7-GHz TB with wind direction. Wind direction variability may be the dominate source of error in the TS retrieval if the TB wind direction signal is large. We are currently studying the TB wind direction effect in considerable detail using a combination of SSM/I, TMI and collocated buoy observations. We originally planed to use the AMSR aboard the ADEOS-2 spacecraft to develop and test the AMSR-E ocean algorithm. Now that the ADEOS-2 launch date has slipped to 2001, this is no longer possible. We are placing more attention on the TMI data set for AMSR al- gorithm development. However, the final specification of the 6.9 GHz emissivity will need to be done after the AMSR-E launch. We expect that the 6.9 GHz emissivity can be relative- ly quickly specified given 1 to 3 months of AMSR observations. 1.5. Concerns Regarding Sea-Surface Temperature Retrieval The capability of measuring sea-surface temperature TS through clouds has long been a goal of microwave radiometry. A global TS product unaffected by clouds and aerosols would be of great benefit to both the scientific and commercial communities. AMSR will be the first satellite sensor to furnish this product, provided that certain requirements are met. 3
  12. 12. Calibrate Radiative Transfer Model (RTM) using SSM/I and SeaSat SMMR observations Development first version of AMSR algorithm using simulated TB's computed by RTM Version 1 Pre-Launch AMSR Algorithm Use TRMM results to specify TRMM parameters at 10.7 GHz Version 2 Pre-Launch AMSR Algorithm Final Prelaunch Testing Final Prelaunch AMSR Algorithm for EOS PM Specify the Sea Surface Emissivity at 6.9 GHz Using Actual On-Orbit AMSR Observations Version 1 Post-Launch AMSR Algorithm Fig. 1. Development steps for ocean algorithm 4
  13. 13. The retrieval of sea-surface temperature to an accuracy of 0.5 K requires the following: 1. Radiometer noise for the 6.9V channel be about 0.1K 2. Incidence angle be known to an accuracy of 0.05° 3. Radio frequency interference (RFI) be less than 0.1 K. 4. The retrieval algorithm be able to separate wind effects from TS effects The first two conditions will be satisfied if the AMSR instrument specifications are met. The radiometer noise figure for one 6.9 GHz observation is 0.3 K. However, the 6.9 GHz observations are greatly over sampled. Observations are taken every 10 km, but the spatial resolution of the footprint is 58 km. During the Level-2A processing, adjacent observations are averaged together in such a way as to reduce the noise to 0.1 K. In doing this averaging, the spatial resolution is degraded by only 2%. The pointing knowledge for the PM platform should be sufficient to meet the incidence angle requirement, as is discussed in Section 3.6. The last two conditions are our major concern. The band from 5.9 to 7.8 GHz is allocated to various communication links. The possibility exists that the sidelobe transmissions from these links will contaminate the AMSR 6.9 GHz measurements. Clearly, this problem needs more attention. A survey of relevant communication links need to be made and sidelobe contamination calculations need to be done. From an algorithm standpoint, the most difficult part of the TS retrieval is separating the TS signal from the wind signal. The T B wind signal is due to both wind speed and wind di- rection variations. It is relatively easy to distinguish wind speed variations from a TS varia- tion. Wind speed mostly affects the h-pol channel and TS mostly affects the v-pol channel. Thus the polarization signature of the observations provides the means to separate T S from W. However, wind direction variations are more problematic in that both polarizations are affected. Simulations (see Section 3) show that without (with) wind direction variability, the TS retrieval error is 0.3 (0.6). These results are contingent on the assumed amplitude for the wind directional TB signal at 6.9 GHz. If the wind direction variation proves to be a domi- nant error source, then we will need to make a correction to the TS retrieval based on some wind direction database, as is discussed in Section 4.3. Note that in contrast to IR retrieval techniques, the atmospheric interference at 6.9 GHz is very small and easily removed using the higher frequency channels, except when there is rain. And, observations affected by rain are easily detected and can be discarded. Thus, the atmosphere does not pose a problem for the TS retrieval. 1.6. Historical Perspective In the 1960’s, it was first recognized that microwave radiometers had the ability to mea- sure atmospheric water vapor V and cloud liquid water L [Barret and Chung, 1962; Staelin; 1966]. In 1972, Nimbus-5 satellite was launched. Aboard Nimbus-5 was the Nimbus-E Mi- crowave Spectrometer (NEMS), which had channels at 22.235 and 31.4 GHz. Staelin et al. [1976] and Grody [1976] demonstrated that water vapor and cloud water could indeed be re- trieved from the NEMS TB’s. In these retrievals they ignored the effect of wind at the ocean surface; at these frequencies the effect of TS is minimal. 5
  14. 14. In the years preceding the launch of Nimbus-5, there were several developments con- cerning the effect of wind at the ocean's surface. Stogryn [1967] developed a theory to ac- count for the wind-induced roughness, and Hollinger [1971] made some radiometric mea- surements from a fixed tower to test the theory. He removed the most obvious foam effects from the data and found that the roughness effect was somewhat less than the Stogryn theory would predict by a frequency dependent factor. Using airborne data, Nordberg et al. [1971] characterized the combined foam and roughness effect at 19.35 GHz. At their measurement angle the observed effect was dominated by foam. Stogryn’s geometric optics theory was extended to included diffraction effects, multiple scattering, and two-scale partitioning by Wu and Fung [1972] and Wentz [1975]. The first simultaneous retrieval of W, V, and L was based on airborne data from the 1973 joint US-USSR Bering Sea Experiment (BESEX) [Wilheit and Fowler, 1977]. Later Chang and Wilheit [1979] combined two NIMBUS-5 instruments, the ESMR and the NEMS for a W, V, and L retrieval. Wilheit [1979a] used the 37-GHz dual polarized data from the Electrically Scanned Microwave Radiomter (ESMR) to explore the wind-induced roughness of the ocean surface. This was later combined with other data to generate a semi-empirical model for the ocean surface emissivity [Wilheit, 1979b] in preparation for the 1978 launch of the Scanning Multichannel Microwave Radiometer (SMMR) on the Nimbus-7 and SeaSat satellites. A theory for the retrieval of all 4 ocean parameters was published by Wilheit and Chang [1980]. The launch of the SeaSat and Nimbus-7 SMMR’s spurred many investigation on SMMR retrieval algorithms and model functions [Wentz, 1983; Njoku and Swanson, 1983; Al- ishouse, 1983; Chang et al, 1984; Gloersen et al., 1984], and the state-of-the-art in oceanic microwave radiometry quickly advanced. It became clear that the water vapor retrievals were highly accurate. A major improvement in the wind retrieval was made when Wentz et al. [1986] combined the SeaSat SMMR TB’s and the SeaSat scatterometer (SASS) wind re- trievals to develop an accurate, semi-empirical relationship for the wind-induced emissivity. Sea-surface temperature retrievals have been less successful. The measurement of TS re- quires relatively low microwave frequencies (4-10 GHz). The SMMR’s were the first satel- lite sensors with the appropriate frequencies to retrieve TS. However, the SMMR’s suffered from a poor calibration design, and the reported T S retrievals [Njoku and Swanson, 1983; Milman and Wilheit, 1985] were useful for little more than a demonstration of the possibility of TS retrievals for future, better calibrated radiometers. The next major milestone was the launch of the Special Sensor Microwave Imager (SSM/I) in 1987. In contrast to SMMR, SSM/I has an external calibration system that pro- vides stable observations. Unfortunately, the lowest SSM/I frequency is 19.3 GHz, and hence TS retrievals are not possible. Shortly after the launch, there was a flurry of new SSM/ I algorithms. Most of these algorithms, such as the Goodberlet et al. [1989] wind algorithm and the Alishouse et al. [1990] vapor algorithm, were simply statistical regressions to in situ data (see Section 3.5). These algorithms performed reasonably well but provided no infor- mation on the relevant physics. A more physical approach to algorithm development for SSM/I was taken by Schluessel and Luthardt [1991] and Wentz [1992, 1997]. This physical approach to algorithm development is described herein and will be the basis for the AMSR ocean algorithm. 6
  15. 15. In November 1997, the first microwave radiometer capable of accurately measuring SST through clouds was launched on the Tropical Rainfall Measuring Mission (TRMM) space- craft. The TRMM Microwave Imager (TMI) is providing an unprecedented view of the oceans. Its lowest frequency channel (10.7 GHz) penetrates non-raining clouds with little at- tenuation, giving a clear view of the sea surface under all weather conditions except rain. Furthermore at this frequency, atmospheric aerosols have no effect, making it possible to produce a very reliable SST time series for climate studies. The one disadvantage of the mi- crowave SST is spatial resolution. The radiation wavelength at 10.7 GHz is about 3 cm, and at these long wavelengths the spatial resolution on the Earth surface for a single TMI obser- vation is about 50 km. Also, the TRMM orbit was selected for continuous monitoring of the tropics. To achieve this, a low inclination angle was chosen, confining the TRMM observa- tions between 40°S and 40°N. Previous microwave radiometers were either too poorly cali- brated or operated at too high of a frequency to provide a reliable estimate of SST. The early results for the TMI SST retrievals are quite impressive and are already leading to improved analyses in a number of important scientific areas, including tropical instability waves (TIWs) and tropical storms [Wentz et al., 1999] 1.7. AMSR Instrument Characteristics The PM AMSR instrument is similar to SSM/I. The major differences are that AMSR has more channels and a larger parabolic reflector. AMSR takes dual polarization observa- tions (v-pol and h-pol) at the 6 frequencies shown in Table 3. The offset 1.6-m diameter parabolic reflector focuses the Earth radiation into an array of 6 feedhorns. The radiation collected by the feedhorns is then amplified by 14 separate total-power radiometers. The 18.7 and 23.8 GHz receivers share a feedhorn, while dedicated feedhorns are provided for the other frequencies. Two feedhorns are required for the 89 GHz channels to achieve a 5- km along-track spacing. Figure 2 shows the block diagram for this configuration. The parabolic reflector and feedhorn array are mounted on a drum containing the ra- diometers, digital data subsystem, mechanical scanning subsystem, and power subsystem. The reflector/feed/drum assembly is rotated about the axis of the drum by a coaxially mount- ed bearing and power transfer assembly. All data, commands, timing and telemetry signals, and power pass through the assembly on slip ring connectors to the rotating assembly. A cold reflector and a warm load are mounted on the transfer assembly shaft and do not rotate with the drum assembly. They are positioned off axis such that they pass between the feedhorn array and the parabolic reflector, occulting it once each scan. The cold reflector re- flects cold sky radiation into the feedhorn array thus serving, along with the warm load, as calibration references for the AMSR. The AMSR rotates continuously about an axis parallel to the spacecraft nadir at 40 rpm. At an altitude of 705 km, it measures the upwelling Earth brightness temperatures over an angular sector of ± 61° degrees about the sub-satellite track, resulting in a swath width of 1445 km. During a scan period of 1.5 seconds, the spacecraft sub-satellite point travels 10 km. Even though the instantaneous field-of-view for each channel is different, Earth obser- vations are recorded at equal intervals of 10 km (5 km for the 89 GHz channels) along the scan. The two 89-GHz feedhorns are offset such that their two scan lines are separated by 5 km in the along-track direction. The nadir angle for the parabolic reflector is fixed at 47.4°, 7
  16. 16. which results in an Earth incidence angle θi of 55° ± 0.3°. The small variation in θi is due to the slight eccentricity of the orbit and the oblateness of the Earth. As compared to the PM AMSR. the AMSR to fly on the ADEOS-2 spacecraft has two additional frequencies: 50.3 and 52.8 GHz. The tables in Section 2 for the radiative transfer model include these two additional frequencies. Table 3. Instrument Specifications for PM AMSR Center Frequencies (GHz) 6.925 10.65 18.7 23.8 36.5 89.0 Bandwidth (MHz) 350 100 200 400 1000 3000 Sensitivity (K) 0.3 0.6 0.6 0.6 0.6 1.1 IFOV (km x km) 76 × 44 49× 28 28 × 16 31 × 18 14 × 8 6×4 Sampling Rate (km x km) 10 × 10 10 × 10 10× 10 10 × 10 10 × 10 5×5 Integration Time (msec) 2.6 2.6 2.6 2.6 2.6 1.3 Main Beam Efficiency (%) 95.3 95.0 96.3 96.4 95.3 96.0 Beamwidth (degrees) 2.2 1.4 0.8 0.9 0.4 0.18 8
  17. 17. 6.9 GHz V-Pol Square-Law 6.9 V counts Receiver Detector 6.9 GHz H-Pol Square-Law 6.9 H counts Receiver Detector 10.7 GHz V-Pol Square-Law 10.7 V counts Receiver Detector 10.7 GHz H-Pol Square-Law 10.7 H counts Receiver Detector 18.7 GHz V-Pol Square-Law 18.7 V counts Receiver Detector 23.8 GHz V-Pol Square-Law Receiver Detector 23.8 V counts 18.7 GHz H-Pol Square-Law 18.7 H counts Receiver Detector 23.8 GHz H-Pol Square-Law 23.8 H counts Receiver Detector 36.5 GHz V-Pol Square-Law Receiver Detector 36.5 V counts 36.5 GHz H-Pol Square-Law 36.5 H counts Receiver Detector 89.0 GHz V-Pol Square-Law 89.0 V counts, A-scan Receiver Detector 89.0 GHz H-Pol Square-Law Receiver Detector 89.0 H counts, A-scan 89.0 GHz V-Pol Square-Law Receiver Detector 89.0 V counts, B-scan 89.0 GHz H-Pol Square-Law 89.0 H counts, B-scan Receiver Detector Fig. 2. Block diagram for PM AMSR feedhorns and radiometers. 9
  18. 18. 2. Geophysical Model for the Ocean and Atmosphere 2.1. Introduction The key component of the ocean parameter retrieval algorithm is the geophysical model for the ocean and atmosphere. It is this model that relates the observed brightness tempera- ture (TB) to the relevant geophysical parameters. In remote sensing, the specification of the geophysical model is sometimes referred to as the forward problem in contrast to the inverse problem of inverting the model to retrieve parameters. An accurate specification of the geo- physical model is the crucial first step in developing the retrieval algorithm. 2.2. Radiative Transfer Equation We begin by deriving the radiative transfer model for the atmosphere bounded on the bottom by the Earth’s surface and on the top by cold space. The derivation is greatly simpli- fied by using the absorption-emission approximation in which radiative scattering from large rain drops and ice particles is not included. Over the spectral range from 6 to 37 GHz, the absorption-emission approximation is valid for clear and cloudy skies and for light rain up to about 2 mm/h. The results of Wentz and Spencer [1997] indicate that only 3% of the SSM/I observations over the oceans viewed rain rates exceeding 2 mm/h. Thus, the absorption- emission model to be presented will be applicable to about 97% of the AMSR ocean obser- vations. In the microwave region, the radiative transfer equation is generally given in terms of the radiation brightness temperature (TB), rather than radiation intensity. So we first give a brief discussion on the relationship between radiation intensity and T B. Let Pλ denote the power within the wavelength range dλ, coming from a surface area dA, and propagating into the solid angle dΩ. The specific intensity of radiation Iλ is then defined by P = I λ cos θi dλ dA dΩ λ (1) The specific intensity is in units of erg/s-cm3-steradian. The angle θi is the incidence angle defined as the angle between the surface normal and the propagation direction. For a black body, Iλ is given by Planck’s law to be [Reif, 1965] 2 hc 2 Iλ = a λ5 exp hc λkT − 1f (2) where c is the speed of light (2.998×1010 cm/s), h is Planck’s constant (6.626×10−27 erg-s), k is Boltzmann’s constant (1.381×10−16 erg/K), λ (cm) is the radiation wavelength, and T (K) is the temperature of the black body. Consider a surface that is emitting radiation with a spe- cific intensity Iλ. Then the brightness temperature TB for this surface is defined as the tem- perature at which a black body would emit the radiation having specific intensity Iλ. In the microwave region, the exponent in (2) is small compared to unity, and (2) can be easily in- verted to give TB in terms of Iν. λ4 I λ TB = (3) 2 kc 10
  19. 19. This approximation is the well known Rayleigh Jeans approximation for λ >> hc/kT. In terms of TB, the differential equation governing the radiative transfer through the at- mosphere is ∂TB = −α(s) TB (s) − T(s) (4) ∂s where the variable s is the distance along some specified path through the atmosphere. The terms α(s) and T(s) are the absorption coefficient and the atmospheric temperature at posi- tion s. Equation (4) is simply stating that the change in T B is due to (1) the absorption of ra- diation arriving at s and (2) to emission of radiation emanating from s. We let s = 0 denote the Earth’s surface, and let s = S denote the top of the atmosphere (i.e., the elevation above which α(s) is essentially zero). Two boundary conditions that correspond to the Earth’s surface at the bottom and cold space at the top are applied to equation (4). The surface boundary conditions states that the upwelling brightness temperature at the surface TB↑ is the sum of the direct surface emission and the downwelling radiation that is scattered upward by the rough surface [Peake, 1959]. z z b g b g π2 2π sec θ i TBAk i ,0) = E(k i )TS + ( sin θs dθs dϕ s TBBks ,0) σ o,c k s , k i + σ o, × k s , ki ( (5) 4π 0 0 where the first TB argument denotes the propagation direction of the radiation and the second argument denotes the path length s. The unit propagation vectors ki and ks denote the direc- tion of the upwelling and downwelling radiation, respectively. In terms of polar angles in a coordinate system having the z-axis normal to the Earth’s surface, these propagation vectors are given by ki = cos ϕ i sin θi ,sin ϕ i sin θi ,cos θi (6a) k s = − cos ϕ s sin θs ,sin ϕ s sin θs , cos θs (6b) The first term in (5) is the emission from the surface, which is the product of the surface temperature TS and the surface emissivity E(ki). The second term is the integral of down- welling radiation TB↓(ks) that is scattered in direction ki. The integral is over the 2π steradian of the upper hemisphere. The rough surface scattering is characterized by the bistatic nor- malized radar cross sections (NRCS) σo,c(θs,θi) and σo,×(θs,θi). These cross sections specify what fraction of power coming from ks is scattered into ki. The subscripts c and × denote co- polarization (i.e., incoming and outgoing polarization are the same) and cross-polarization (i.e., incoming and outgoing polarizations are orthogonal), respectively. The cross sections also determine the surface reflectivity R(ki) via the following integral. z z b g b g π2 2π sec θi R( k i ) = sin θs dθs dϕ s σ o,c k s , ki + σ o, × k s , k i (7) 4π 0 0 The surface emissivity E(ki) is given by Kirchhoff’s law to be E( k i ) = 1 − R( k i ) (8) 11
  20. 20. It is important to note that equations (5) an (7) provide the link between passive microwave radiometry and active microwave scatterometry. The scatterometer measures the radar backscatter coefficient, which is simply σo,c(-ki,ki). The upper boundary condition for cold space is TBBk s , S) = TC ( (9) This simply states that the radiation coming from cold space is isotropic and has a magnitude of TC = 2.7 K. The differential equation (4) is readily solved by integrating and applying the two bound- ary conditions (5) and (9). The result for the upwelling brightness temperature at the top of the atmosphere (i.e., the value observed by Earth-orbiting satellites) is TBAk i , S) = TBU + τ ETS + TBΩ ( (10) where TBU is the contribution of the upwelling atmospheric emission, τ is the total transmit- tance from the surface to the top of the atmosphere, E is the Earth surface emissivity given by (8), and TBΩ is the surface scattering integral in (5). The atmospheric terms can be ex- pressed in terms of the transmittance function τ(s1,s2) b g Fz I G J s2 H K τ s1 , s2 = exp − ds α(s) (11) s1 which is the transmittance between points s1 and s2 along the propagation path ks or ki. The total transmittance τ in (10) is given by τ = τ 0,Saf (12) and the upwelling and downwelling atmosphere emissions are given by z S TBU = ds α(s) T(s) τ(s, S) (13a) z 0 S TBD = ds α(s) T(s) τ(0, s) (13b) 0 The sky radiation scattering integral is z z b g b g π2 2π sec θ i TBΩ = sin θs dθs dϕ s ( TBD + τTC ) σ o,c k s , k i + σ o, × k s , ki (14) 4π 0 0 Thus, given the temperature TS and absorption coefficient α at all points in the atmo- sphere and given the surface bistatic cross sections, T B can be rigorously calculated. Howev- er, in practice, the 3-dimensional specification of TS and α over the entire volume of the at- mosphere is not feasible, and to simplify the problem, the assumption of horizontal uniformi- ty is made. That is to say, the absorption is assumed to only be a function of the altitude h above the Earth’s surface, i.e., α(s) = α(h). To change the integration variable from ds to dh, we note that for the spherical Earth 12
  21. 21. ∂s 1+ δ = (15) ∂h cos θ + δ(2 + δ ) 2 where θ is either θi or θs, δ = h/RE, and RE is the radius of the Earth. In the troposphere δ << 1, and an excellent approximation for θ < 60° is, ∂s = sec θ (16) ∂h With this approximation and the assumption of horizontal uniformity, the above equations reduce to the following expressions. Fsec dh I τb , h , θg expG θ z α( h )J h2 H K h 1 2 = − (17) h1 τ = τb H, θ g 0, (18) z i H TBU = sec θi dh α( h ) T( h ) τ( h, H, θi ) (19a) z 0 H TBD = sec θs dh α( h ) T( h ) τ(0, h, θs ) (19b) 0 Thus, the brightness temperature computation now only requires the vertical profiles of T(h) and α(h) along with the surface cross sections. The following two sections discuss the atmo- spheric model for α(h) and the sea-surface model for the cross sections, respectively. In closing, we note that the AMSR incidence angle is 55° and hence approximation (16) is quite valid, with one exception. In the scattering integral, θs goes out to 90°, and in this case we use (15) to evaluate the integral. 2.3. Model for the Atmosphere In the microwave spectrum below 100 GHz, atmospheric absorption is due to three com- ponents: oxygen, water vapor, and liquid water in the form of clouds and rain [Waters, 1976]. The sum of these three components gives the total absorption coefficient (napers/cm). α( h ) = α O ( h ) + α V ( h ) + α L ( h ) (20) Numerous investigators have studied the dependence of the oxygen and water vapor coeffi- cients on frequency ν (GHz), temperature T (K), pressure P (mb), and water vapor density ρV (g/cm3) [Becker and Autler, 1946; Rozenkranz, 1975; Waters, 1976; Liebe, 1985]. To specify αO and αV as a function of (ν,T,P,ρV) we use the Liebe [1985] expressions with one modification. The self-broadening component of the water vapor continuum is reduced by a factor of 0.52 (see below). The liquid water coefficient αL comes directly from the Rayleigh approximation to Mie scattering and is a function of T and the liquid water density ρL (g/cm2) (see below). Figure 3 shows the total atmospheric absorption for each component. Results for three water vapor cases (10, 30, and 60 mm) are shown. The cloud water content is 0.2 mm. This corresponds to a moderately heavy non-raining cloud layer. 13
  22. 22. Let AI denote the vertically integrated absorption coefficient. z H A I = dh α I ( h ) (21) 0 where h is the height (cm) above the Earth’s surface and subscript I equals O, V, or L. Equa- tions (17) and (18) then give the total transmittance to be b τ = exp − sec θ i A O + A V + A L g (22) Assuming for the moment that the atmospheric temperature is constant, i.e., T(h) = T, then the integrals in equations (19) can be exactly evaluated in closed form to yield a f TBU = TBD = 1 − τ T (23) In reality, the atmospheric temperature does vary with h, typically decreasing at a lapse rate of about -5.5 C/km in the lower to mid troposphere. In view of (23), we find it convenient to parameterize the atmospheric model in terms of the following upwelling and downwelling effective air temperatures: TU = TBU / (1 − τ) (24a) TD = TBD / (1 − τ) (24b) These effective temperatures are indicative of the air temperature averaged over the lower to mid troposphere. Note that in the absence of significant rain, TU and TD are very similar in value, with TU being 1 to 2 K colder. In view of the above equations, one sees that the atmospheric model can be parameter- ized in terms of the following 5 parameters: 1. Upwelling effective temperature TU 2. Downwelling effective temperature TD 3. Vertically integrated oxygen absorption AO 4. Vertically integrated water vapor absorption AV 5. Vertically integrated liquid water absorption AL To study the properties of the first four parameters, we use a large set of 42,195 radiosonde flights launched from small islands [Wentz, 1997]. These radiosonde reports provide air temperature T(h), air pressure p(h), and water vapor density ρV(h) at a number of levels in the troposphere. From these data, the coefficients αO and αV are computed from the Liebe [1985] expressions, except that the water vapor continuum term is modified as discussed in the next paragraph. Performing the numerical integrations as indicated above, T U, TD, AO, and AV are found for each radiosonde flight. In addition, the vertically integrated water va- por V is also computed. 14
  23. 23. 100. 10. Total Atmosphere Attenuation (naper) 1.0 0.1 0.01 Legend Oxygen Water Vapor, 10 mm Water Vapor, 30 mm 0.001 Water Vapor, 60 mm Cloud, 0.20 mm 0.0001 0 10 20 30 40 50 60 70 80 90 100 Frequency (GHz) Fig. 3. The atmospheric absorption spectrum for oxygen, water vapor, and cloud water. Results for three water vapor cases (10, 30, and 60 mm) are shown. The cloud water content is 0.2 mm which corresponds to a moderately heavy non-raining cloud layer. 15
  24. 24. z H V = 10 dh ρV ( h ) (25) 0 where ρV(h) is in units of g/cm3, and the leading factor of 10 converts from g/cm2 to mm. Wentz [1997] computed AV directly from collocated SSM/I and radiosonde observations. At 19, 22, and 37 GHz, the Liebe AV was found to be 4%, 3%, and 20% higher than the SSM/I-derived value, respectively. To quote Liebe [1985]: ‘Water vapor continuum absorp- tion has been a major source of uncertainty in predicting millimeter wave attenuation rates, especially in the window ranges.’ The frequency of 37 GHz is in a water vapor window and is most affected by the continuum. It should be noted that Liebe also needed to rely on com- bined radiometer-radiosonde measurements to infer the continuum in the 6 to 37 GHz re- gion. Liebe’s data set in this spectral region is rather limited and does not contain any 37 GHz observations. We believe the SSM/I method of deriving A V is more accurate than Liebe’s method, and hence adjust the Liebe [1985] water vapor spectrum so that it will agree with the SSM/I results. We find that very good agreement is obtained by reducing the self- broadening component of the water vapor continuum by a factor of 0.52. After this adjust- ment, the agreement at all three frequencies is within ± 1%. Figure 4 shows the TD values computed from the 42,195 radiosondes plotted versus V. Three frequencies are shown (19, 22, and 37 GHz), and the curves are quite similar. The solid lines in the figure show equation (26), and vertical bars show the ± one standard devia- tion of TD derived from the radiosondes. For low to moderate values of V (0 to 40 mm), T D increases with V, and above 40 mm, TD reaches a relatively constant value of 287 K. The T U versus V curves (not shown) are very similar except that T U is 1 to 2 K colder. The follow- ing least-square regressions are found to be a good approximation of the TD, TU versus V re- lationship: b TD = b 0 + b1V + b 2V 2 + b3V3 + b 4V 4 + b5 ς TS − TV g (26a) TU = TD + b 6 + b 7 V (26b) where TV = 27316 + 0.8337 V − 3.029 × 10 −5 V3.33 . V ≤ 48 (27a) TV = 30116 . V > 48 (27b) and b gF (T1200 ) I |T − T | ≤ 20K G −T J 2 H ς (TS − TV ) = 1.05 ⋅ TS − TV ⋅ 1 − K S V S V (27c) ς (TS − TV ) = sign(TS − TV ) ⋅14K |TS − TV |> 20K (27d) V is in units of millimeters and all temperatures are in units of Kelvin. When evaluating (26a), the expression is linearly extrapolated when V is greater than 58 mm. We have in- cluded a small additional term that is a function of the difference between the sea-surface temperature TS and TV, which represents the sea-surface temperature that is typical for water vapor V. The term ς ( TS − TV ) accounts for the fact that the effective air temperature is typi 16
  25. 25. 290 280 Effective Air Temperature for Downwelling Radiation (K) 270 260 250 19 GHz 240 290 280 270 260 250 22 GHz 240 290 280 270 260 250 37 GHz 240 0 10 20 30 40 50 60 70 Columnar Water Vapor (mm) Fig. 4. The effective air temperature TD for downwelling radiation plotted versus the RAOB columnar water vapor. The solid curve is the model value, and the vertical bars are the ± one standard deviation of TD derived from radiosondes. cally higher (lower) for the case of unusually warm (cold) water. The TV versus V relation- ship was obtained by regressing the climatology sea-surface temperature at the radiosonde site to V derived from the radiosondes. Over the full range of V, the rms error in approxi- 17
  26. 26. mation (26) is typically about 3 K. Table 4 gives the b0 through b7 coefficients for all 8 AMSR frequencies. Table 4. Model Coefficients for the Atmosphere Freq. (GHz) 6.93E+0 10.65E+0 18.70E+0 23.80E+0 36.50E+0 50.30E+0 52.80E+0 89.00E+0 b0 (K) 239.50E+0 239.51E+0 240.24E+0 241.69E+0 239.45E+0 242.10E+0 245.87E+0 242.58E+0 b1 (K mm−1) 213.92E−2 225.19E−2 298.88E−2 310.32E−2 254.41E−2 229.17E−2 250.61E−2 302.33E−2 b2 (K mm−2) −460.60E− −446.86E− −725.93E− −814.29E− −512.84E− −508.05E− −627.89E− −749.76E−4 4 4 4 4 4 4 4 b3 (K mm−3) 457.11E−6 391.82E−6 814.50E−6 998.93E−6 452.02E−6 536.90E−6 759.62E−6 880.66E−6 b4 (K mm−4) −16.84E−7 −12.20E−7 −36.07E−7 −48.37E−7 −14.36E−7 −22.07E−7 −36.06E−7 −40.88E−7 b5 0.50E+0 0.54E+0 0.61E+0 0.20E+0 0.58E+0 0.52E+0 0.53E+0 0.62E+0 b6 (K) −0.11E+0 −0.12E+0 −0.16E+0 −0.20E+0 −0.57E+0 −4.59E+0 −12.52E+0 −0.57E+0 b7 (K mm−1) −0.21E−2 −0.34E−2 −1.69E−2 −5.21E−2 −2.38E−2 −8.78E−2 −23.26E−2 −8.07E−2 aO1 8.34E−3 9.08E−3 12.15E−3 15.75E−3 40.06E−3 353.72E−3 1131.76E−3 53.35E−3 −1 aO2 (K ) −0.48E−4 −0.47E−4 −0.61E−4 −0.87E−4 −2.00E−4 −13.79E−4 −2.26E−4 −1.18E−4 aV1 (mm−1) 0.07E−3 0.18E−3 1.73E−3 5.14E−3 1.88E−3 2.91E−3 3.17E−3 8.78E−3 −2 aV2 (mm ) 0.00E−5 0.00E−5 −0.05E−5 0.19E−5 0.09E−5 0.24E−5 0.27E−5 0.80E−5 Table 5. RMS Error in Oxygen and Water Vapor Absorption Approximation Freq. (GHz) 6.93 10.65 18.70 23.80 36.50 50.30 52.80 89.00 Oxygen, AO 0.0002 0.0002 0.0003 0.0003 0.0008 0.0062 0.0163 0.0009 Vapor, AV 0.0001 0.0002 0.0011 0.0013 0.0025 0.0042 0.0046 0.0129 The vertically integrated oxygen absorption AO is nearly constant over the globe, with a small dependence on the air temperature. We find the following expression to be a very good approximation for AO: b A O = a O1 + a O 2 TD − 270 g (28) Table 4 gives the aO coefficients for the 8 AMSR frequencies, and Table 5 gives the rms er- ror in this approximation for the 8 frequencies. At 23.8 GHz and below, the error is negligi- ble, being 0.0003 napers or less. At 36.5 GHz, the error is still quite small, being 0.0008 napers. Note that 0.001 napers roughly corresponds to a T B error of 0.5 K. For the 50.3 and 52.8 GHz oxygen band channels, the error is considerably larger, but (28) is not used for the oxygen band channels. Rather the oxygen band channels can be used to retrieve TD. The vapor absorption AV is primarily a linear function of V, although there is a small sec- ond order term. We find the following expression is a good approximation for AV: AV = aV1V + aV2V2 (29) Table 4 gives the aV coefficients for the 8 AMSR frequencies, and Table 5 gives the rms er- ror in this approximation for the 8 frequencies. For the 6.9 and 10.7 AMSR channels, the rms error in this approximation is negligible, being 0.0002 napers or less. In the 18.7 to 36.5 range, the error remains relatively small (0.001 to 0.0025 napers), but not negligible. 18
  27. 27. The final atmospheric parameter to be specified is the vertically integrated liquid water absorption AL. When the liquid water drop radius is small relative to the radiation wave- length, the absorption coefficient αL (cm−1) is given by the Rayleigh scattering approxima- tion [Goldstein, 1951]: 6 πρL ( h ) F I 1− ε αL = λρo H K Im 2+ε (30) where λ is the radiation wavelength (cm), ρL(h) is the density (g/cm3) of cloud water in the atmosphere given as a function of h, ρo is the density of water (ρo ≈ 1 g/cm3), and ε is the complex dielectric constant of water. Note that the dielectric constant varies with tempera- ture and hence is also a function of h. Substituting (30) into (21) gives 0.6 πL F I 1− ε AL = λ Im 2+εH K (31) where L is the vertically integrated liquid water (mm) given by z H L = 10 dh ρL ( h ) (32) 0 The leading factor of 10 converts from g/cm2 to mm. In deriving (31), we have assumed the cloud is at a constant temperature. For the more realistic case of the temperature varying with height, ε should be evaluated at some mean effective temperature for the cloud. The specification of ε as a function of temperature and frequency is given in Section 2.4. An ex- cellent approximation for (31) is found to be A L = a L1 1 − a L 2 (TL − 283) L (33) where TL is the mean temperature of the cloud, and the aL coefficients are given in Table 6 for the 8 AMSR frequencies. The error in this approximation is ≤ 1% over the range of TL from 273 to 288 K, which is negligible compared to other errors such as the uncertainty in specifying the cloud temperature TL. Note that in the retrieval algorithm, the error in speci- fying TL only effects the retrieved value of L. The retrieval of the other parameters only re- quires the spectral ratio of AL, which is essentially independent of T L due to the fact that aL2 is spectrally flat. In the absence of rain, the cloud droplets are much smaller than the radiation wave- lengths being considered, and equations (31) and (33) are valid. When rain is present, Mie scattering theory must be used to compute AL. For light rain not exceeding 2 mm/h and for frequencies between 6 and 37 GHz, the Mie scattering computations give the following ap- proximation [Wentz and Spencer, 1998]: 19
  28. 28. Table 6. Coefficients for Rayleigh Absorption and Mie Scattering. Freq (GHz) 6.93 10.65 18.70 23.80 36.50 50.30 52.80 89.00 aL1 0.0078 0.0183 0.0556 0.0891 0.2027 0.3682 0.4021 0.9693 aL2 0.0303 0.0298 0.0288 0.0281 0.0261 0.0236 0.0231 0.0146 aL3 0.0007 0.0027 0.0113 0.0188 0.0425 0.0731 0.0786 0.1506 aL4 0.0000 0.0060 0.0040 0.0020 -0.0020 -0.0020 -0.0020 -0.0020 aL5 1.2216 1.1795 1.0636 1.0220 0.9546 0.8983 0.8943 0.7961 A R = a L3 ⋅ 1 + a L4 ⋅ ( TL − 283) ⋅ H ⋅ R a L5 (34a) The rain column height H (in km) can be approximated by: H = 1+ 0.14 ⋅ (TS − 273) − 0.0025 ⋅ (TS − 273)2 if TS < 301 (34b) H = 2.96 if TS ≥ 301, (34c) where TS denotes the sea surface temperature (in K). The rain rate R (in mm/h) is related to the liquid cloud water density L by d L = 0.18 ⋅ 1 + HR . i (34d) In deriving (34a) we have used a Marshall and Palmer [1948] drop size distribution. 2.4. Dielectric Constant of Sea-Water and the Specular Sea Surface A key component of the sea-surface model is the dielectric constant ε of sea water. The parameter is a complex number that depends on frequency ν, water temperature TS, and wa- ter salinity s. The dielectric constant is given by [Debye,1929; Cole and Cole, 1941] as εS − ε∞ 2 jσλ ε = ε∞ + 1− η − (35) 1 + jλ R λ c where j = −1 , λ (cm) is the radiation wavelength, εε is the dielectric constant at infinite frequency, εS is the dielectric constant for zero frequency (i.e., the static dielectric constant), and λR (cm) is the relaxation wavelength. The spread factor η is an empirical parameter that describes the distribution of relaxation wavelengths. The last term accounts for the conduc- tivity of salt water. In this term, σ (sec−1, Gaussian units) is the ionic conductivity and c is the speed of light. Several investigators have developed models for the dielectric constant of sea water. In the Stogryn [1971] model the salinity dependence of εS and λR was based on the Lane and Saxton [1952] laboratory measurements of saline solutions. Stogryn noted that the Lane- Saxton measurements for distilled water did not agree with those of other investigators. The Klein and Swift [1977] model is very similar to Stogryn model except that the salinity depen- dence of εS was based on more recent 1.4 GHz measurements [Ho and Hall, 1973; Ho et al., 1974]. Klein-Swift noted that their εS was significantly different from that derived from the 20
  29. 29. Lane and Saxton measurements. It appears that there may be a problem with Lane-Saxton measurements. However, in the Klein-Swift model, the salinity dependence of λR was still based on the Lane-Saxton measurements. We analyzed all the measurements used by Stogryn and Klein-Swift and concluded that the Lane-Saxton measurements of ε for both distilled water and salt water were inconsistent with the results reported by all other investi- gators. Therefore, we completely exclude the Lane-Saxton measurements from our model derivation. The model to be presented is very similar to the Klein-Swift model, with two exceptions. First, since we excluded Lane-Saxton measurements, the salinity dependence of λR is differ- ent. For cold water (0 to 10 C), our λR is about 5% lower than the Klein-Swift value and for warm water (30 C), it is about 1% higher. Second, our value for εε is 4.44 and the Klein- Swift value is 4.9, which was the value used by Stogryn. In the Stogryn model, η = 0, whereas in the Klein-Swift model, η = 0.02. Grant et al. [1957] pointed out that the choice of εε depends on the choice for η, where η = 0 → εε = 4.9 and η = 0.02 → εε = 4.5. Thus the Klein-Swift value of εε = 4.9 is probably too high. In terms of brightness temperatures, these λR and εε differences are most significant at the higher frequencies. For example, at 37 GHz and θi = 55°, the difference in specular brightness temperatures produced by our model and the Klein-Swift model differ by about ± 2 K. Analyses of SSM/I observations show that our new model, as compared to the Klein-Swift model, produces more consistent retrievals of ocean parameters. For example, using the Klein-Swift model resulted in an abundance of negative cloud water retrievals in cold water. This problem no longer occurs with the new model. (The negative cloud water problem was the original motivation for doing this reanal- ysis of the ε model.) We first describe the dielectric constant model for distilled water, and then extend the model to the more general case of a saline solution. The static dielectric constant εS0 for dis- tilled water has been measured by many investigators. The more recent measurements [Malmberg and Maryott, 1956; Archer and Wang, 1990] are in very good agreement (0.2%). The Archer and Wang [1990] values for εS0, which are reported in the Handbook of Chem- istry and Physics [Lide,1993], are regressed to the following expression: ε S0 = 87.90 exp( −0.004585 t S ) (36) where tS is the water temperature in Celsius units. The accuracy of the regression relative to the point values for εS0 is 0.01% over the range from 0 to 40 C. The other three parameters for the dielectric constant of distilled water are the relaxation wavelength λR0, the spread factor η, and εε . We determine these parameters by a least- squares fit of (35) to laboratory measurements εmea of the dielectric constant for the range from 1 to 40 GHz. A literature search yielded ten papers reporting εmea for distilled water. Values for λR0, η, and εε are found so as to minimize the following quantity: Q = Re(ε − ε mea ) + Im(ε − ε mea ) 2 2 (37) The relaxation wavelength is a function of temperature [Grant et al., 1957], but it is general- ly assumed that η and εε are independent of temperature. The least squares fit yields η = 0.012, εε = 4.44, and 21
  30. 30. λ R 0 = 3.30 exp( −0.0346 t S + 0.00017 t S ) 2 (38) These values are in good agreement with those obtained by other investigators. Our λR0 agrees with the expression derived by Stogryn [1971] to within 1%. The values for η (εε ) re- ported in the literature vary from 0 to 0.02 (4 to 5). Note that using a larger value for η ne- cessitates using a smaller value for εε . The presence of salt in the water produces ionic conductivity σ and modifies εS and λR. It is generally assumed that η and εε are not affected by salinity. Weyl [1964] found the fol- lowing regression for the conductivity of sea water. b g σ = 3.39 × 10 9 C 0.892 exp − ∆ tζ (39) c h ζ = 2.03 × 10 −2 + 1.27 × 10 −4 ∆ t + 2.46 × 10 −6 ∆2t − C 3.34 × 10 −5 − 4.60 × 10 −7 ∆ t + 4.60 × 10 −8 ∆2t (40) C = 0.5536 s (41) ∆ t = 25 − t S (42) where s and C are salinity and chlorinity in units of parts/thousand. Note that we have con- verted the Weyl conductivity to Gaussian units of sec−1. To determine the effect of salinity on εS, we use low frequency (1.43 and 2.65 GHz) measurements of ε for sea water and saline solutions [Ho and Hall, 1973; Ho et al., 1974]. For the Ho-Hall data, only the real part of the dielectric constant is used in the fit. Klein and Swift reported that the measurements of the imaginary part were in error. To determine the effect of salinity on λR, we use higher frequency (3 to 24 GHz) measurements of ε for saline solutions [Haggis et al., 1952; Hasted and Sabeh, 1953; Hasted and Roderick, 1958]. A least-squares fit to these data shows that the salinity dependence of εS and λR can be modeled as c ε S = ε S0 exp −3.45 × 10 −3 s + 4.69 × 10 −6 s2 + 1.36 × 10 −5 st S h (43) c h λ R = λ R 0 − 6.54 × 10 −3 1 − 3.06 × 10 −2 t S + 2.0 × 10 −4 t S s 2 (44) The accuracy of the dielectric constant model is characterized in terms of its correspond- ing specular brightness temperature TB. For each laboratory measurement of ε, we compute the specular TB for an incidence angle of 55° using the Fresnel equation (45) below. Two TB’s are computed: one using εmea and the other using the model ε coming from the above equations. For the low frequency Ho-Hall data, the rms difference between the ‘measure- ment’ TB and the ‘model’ TB is about 0.1 K for v-pol and 0.2 K for h-pol. For the higher fre- quency data set, the rms difference is 0.8 K for v-pol and 0.5 K for h-pol. Once the dielectric constant is known, the v-pol and h-pol reflectivity coefficients ρV and ρH for a specular (i.e., perfectly flat) sea surface are calculated from the well-known Fresnel equations ε cos θi − ε − sin 2 θi ρv = (45a) ε cos θi + ε − sin 2 θi 22
  31. 31. cos θi − ε − sin 2 θ i ρh = (45b) cos θ i + ε − sin 2 θi where θI is the incidence angle. The power reflectivity R is then given by 2 R 0 p = ρp (46) where subscript 0 denotes that this is the specular reflectivity and subscript p denotes polar- ization. An analysis using TMI data indicates small deviations from the model function for the di- electric constant of sea water as discussed above. The effect is mainly noted in the v-pol re- flectivity. In order to account for these small differences a correction term of ∆R 0v b g = 4.887 ⋅ 10−8 − 6108 ⋅10−8 ⋅ TS − 273 . 3 is added to the v-pol reflectivity R 0v in (46). The resulting changes in the brightness temper- ature range from about +0.14K in cold water to about –0.36K in warm water. 2.5. The Wind-Roughened Sea Surface It is well known that the microwave emission from the ocean depends on surface rough- ness. A calm sea surface is characterized by a highly polarized emission. When the surface becomes rough, the emission increases and becomes less polarized (except at incidence an- gles above 55º for which the vertically polarized emission decreases). There are three mech- anisms that are responsible for this variation in the emissivity. First, surface waves with wavelengths that are long compared to the radiation wavelength mix the horizontal and verti- cal polarization states and change the local incidence angle. This phenomenon can be mod- eled as a collection of tilted facets, each acting as an independent specular surface [Stogryn, 1967]. The second mechanism is sea foam. This mixture of air and water increases the emissivity for both polarizations. Sea foam models have been developed by Stogryn [1972] and Smith [1988]. The third roughness effect is the diffraction of microwaves by surface waves that are small compared to the radiation wavelength. Rice [1951] provided the basic formulation for computing the scattering from a slightly rough surface. Wu and Fung [1972] and Wentz [1975] applied this scattering formulation to the problem of computing the emis- sivity of a wind-roughened sea surface. These three effects can be parameterized in terms of the rms slope of the large-scale roughness, the fractional foam coverage, and the rms height of the small-scale waves. Each of these parameters depends on wind speed. Cox and Munk [1954], Monahan and O'Muirc- heartaigh [1980], and Mitsuyasu and Honda [1982] derived wind speed relationships for the three parameters, respectively. These wind speed relationships in conjunction with the tilt+foam+diffraction model provide the means to compute the sea-surface emissivity. Com- putations of this type have been done by Wentz [1975, 1983] and are in general agreement with microwave observations. In addition to depending on wind speed, the large-scale rms slope and the small-scale rms height depend on wind direction. The probability density function of the sea-surface slope is skewed in the alongwind axis and has a larger alongwind variance than crosswind 23
  32. 32. variance [Cox and Munk, 1954]. The rms height of capillary waves is very anisotropic [Mit- suyasu and Honda, 1982]. The capillary waves traveling in the alongwind direction have a greater amplitude than those traveling in the crosswind direction. Another type of direction- al dependence occurs because the foam and capillary waves are not uniformly distributed over the underlying structure of large-scale waves. Smith's [1988] aircraft radiometer mea- surements show that the forward plunging side of a breaking wave exhibits distinctly warmer microwave emissions than does the back side. In addition, the capillary waves tend to clus- ter on the downwind side of the larger gravity waves [Cox, 1958; Keller and Wright, 1975]. The dependence of foam and capillary waves on the underlying structure produces an up- wind-downwind asymmetry in the sea-surface emissivity. The anisotropy of capillary waves is responsible for the observed dependence of radar backscattering on wind direction [Jones et al., 1977]. The upwind radar return is consider- ably higher than the crosswind return. Also, the modulation of the capillary waves by the underlying gravity waves causes the upwind return to be generally higher than the downwind return. These directional characteristics of the radar return have provided the means to sense wind direction from aircraft and satellite scatterometers [Jones et al., 1979]. To model the rough sea surface, we begin by assuming the surface can be partitioned into foam-free areas and foam-covered areas within the radiometer footprint. The fraction of the total area that is covered by foam is denoted by f. The composite reflectivity is then given by R = (1 − f )R clear + fκR clear (47) where Rclear is the reflectivity of the rough sea surface clear of foam, and the factor κ ac- counts for the way in which foam modifies the reflectivity. As discussed above, foam tends to decrease the reflectivity, and hence κ < 1. The reflectivity of the clear, rough sea surface is modeled by the following equation: R clear = (1 − β) R geo (48) where Rgeo is the reflectivity given by the standard geometric optics model (see below) and the factor 1 − β accounts for the way in which diffraction modifies the geometric-optics re- flectivity. Wentz [1975] showed that the inclusion of diffraction effects is a relatively small effect and hence β small compared to unity. Combining the above two equations gives R = (1 − F )R geo (49) F = f + β − fβ − fκ + fκβ (50) where F is a ‘catch-all’ term that accounts for both foam and diffraction effects. All of the terms that makeup F are small compared to unity, and the results to be presented show that F < 10%. The reason we lump foam and diffraction effects together is that they both are diffi- cult to model theoretically. Hence, rather than trying to compute F theoretically, we let F be a model parameter that is derived empirically from various radiometer experiments. Howev- er, the Rgeo term is theoretically computed from the geometric optics. Thus, the F term is a measure of that portion of the wind-induced reflectivity that is not explained by the geomet- ric optics. 24
  33. 33. The geometric optics model assumes the surface is represented by a collection of tilted facets, each acting as an independent reflector. The distribution of facets is statistically char- acterized in terms of the probability density function P(S u,Sc) for the slope of the facets, where Su and Sc are the upwind and crosswind slopes respectively. Given this model, the re- flectivity can be computed from equation (7). To do this, the integration variables θs,φs in (7) are transformed to the surface slope variables. The two equations governing this trans- formation are b g k s = k i − 2 ki ⋅ n n (51) − Su , − Sc , 1 n= (52) 1 + S2 + S2 u c where n is the unit normal vector for a given facet. Transforming (7) to the S u,Sc integration zz variables yields b g g b g b 2 z z P(S ,S ) 1 + S + S b ⋅ ng dSu dSc P(Su , Sc ) 1 + S2 + S2 k i ⋅ n p ⋅ hi ρh hs + p ⋅ v i ρv v s χ(k s )(1 − R × ) + R × R geo = u c 2 2 (53) dS dS u c u kc u c i where p is the unit vector specifying the reflectivity polarization. The unit vectors hi and vi (hs and vs) are the horizontal and vertical polarization vectors associated with the propagation vector ki (ks) as measured in the tilted facet reference frame. These polarization vectors in the tilted frame of reference are given by k ×n hj = j (54a) kj × n vj = kj × hj (54b) where subscript j = i or s. The terms ρv and ρh are the v-pol and h-pol Fresnel reflection co- efficients given above. The last factor in (53) accounts for multiple reflection (i.e., radiation reflecting off of one facet and then intersecting another). χ(ks) is the shadowing function given by Wentz [1975], and R× is the reflectivity of the secondary intersection. The shadow- ing function χ(ks) essentially equals unity except when ks approaches surface grazing angles. The interpretation of (53) is straightforward. The integration is over the ensemble of tilt- 2 2 b g ed facets having a slope probability of P(Su,Sc). The term 1 + Su + Sc k i ⋅ n is proportional to the solid angle subtended by the tilted facet as seen from the observation direction speci- b g b g 2 fied by ki. The term p ⋅ hi ρh h s + p ⋅ v i ρv v s is the reflectivity of the tilted facet. And, the denominator in (53) properly normalizes the integral. To specify the slope probability we use a Gaussian distribution as suggested by Cox and Munk [1954], and we assume that the upwind and crosswind slope variances are the same. Wind direction effects are considered in Section 2.7. Then, the slope probability is given by LS − S O P(S , S ) = c S hexp M N∆S P − 2 −1 2 2 π∆ Q (55) u c u c 2 25
  34. 34. where ∆S2 is the total slope variance defined as the sum of the upwind and crosswind slope variances. Ocean waves with wavelengths shorter than the radiation wavelength do not con- tribute to the tilting of facets and hence should not be included in the ensemble specified by P(Su,Sc). For this reason, the effective slope variance ∆S2 increases with frequency, reaching a maximum value referred to as the optical limit. The results of Wilheit and Chang [1980] and Wentz [1983] indicate that the optical limit is reached near ν = 37 GHz. Hence, for ν ≥ 37 GHz, we use the Cox and Munk [1954] expression for optical slope variance. For lower frequencies, a reduction factor is applied to the Cox and Munk expression. This reduction factor is based on ∆S2 values derived from the SeaSat SMMR observations [Wentz, 1983]. ∆S2 = 5.22 × 10 −3 W ν ≥ 37 GHz (56a) ∆S2 = 5.22 × 10 −3 1 − 0.00748(37 − ν)1.3 W ν < 37 GHz (56b) where W is the wind speed (m/s) measured 10 m above the surface. Note the Cox and Munk wind speed was measured at a 12.5 m elevation. Hence, their coefficient of 5.12×10−3 is in- creased by 2% to account for our wind being referenced to a 10 m elevation. The sea-surface reflectivity Rgeo is computed for a range of winds varying from 0 to 20 m/s, for a range of sea-surface temperatures varying from 273 to 303 K, and for a range of incidence angles varying from 49° to 57°. These computations require the numerical evalua- tion of the integral in equation (53). The integration is done over the range S2 + S2 ≤ 4.5∆S2 . u c Facets with slopes exceeding this range contribute little to the integral, and it is not clear if a Gaussian slope distribution is even applicable for such large slopes. Analysis shows that the computed ensemble of Rgeo is well approximated by the following regression: b gb gb g b R geo = R 0 − r0 + r1 θi − 53 + r2 TS − 288 + r3 θi − 53 TS − 288 W g (57) where the first term R0 is the specular power reflectivity given by (46) and the second term is the wind-induced component of the sea-surface reflectivity. The r coefficients are given in Table 7 for all AMSR channels. Equation (57) is valid over the incidence angle from 49° to 57°. It approximates the θi and TS variation of Rgeo with an equivalent accuracy of 0.1 K. The approximation error in the wind dependence is larger. In the geometric optics computa- tions, the variation of Rgeo with wind is not exactly linear. In terms of T B, the non-linear component of Rgeo is about 0.1 K at the lower frequencies and 0.5 K at the higher frequen- cies. However, in view of the general uncertainty in the geometric optics model, we will use the simple linear expression for Rgeo, and let the empirical F term account for any residual non-linear wind variations, as is discussed in the next paragraph. In the case of the coefficients r2 we do not use the geometric optics model coefficients (Ta- ble 7) but rather use the following empirically derived forms (units are s/m-K): bg = −21⋅10 r . 2 v-pol −5 (58) bg = −5.5 ⋅10 r 2 h-pol −5 b g + 0.989 ⋅10−6 ⋅ 37 − ν if ν ≤ 37 (59a) bg = −5.5 ⋅10 r2 h-pol −5 if ν > 37. (59b) This accounts for the observations that the wind induced emissivity is less in warm water. This effect was observed during the monsoons in the Arabian sea. 26